Answer
Answers to a, b, c, d, are all
$\left[\begin{array}{ll}
0.2 & 0.8\\
0.2 & 0.8
\end{array}\right]$
Work Step by Step
$a.$
$P^{2}=\left(\begin{array}{ll}{0.2}&{0.8}\\{0.2}&{0.8}\end{array}\right)\left(\begin{array}{ll}{0.2}&{0.8}\\{0.2}&{0.8}\end{array}\right)=$
$= \left[\begin{array}{ll}
0.2\cdot 0.2+0.8\cdot 0.2 & 0.2\cdot 0.8+0.8\cdot 0.8\\
0.2\cdot 0.2+0.8\cdot 0.2 & 0.2\cdot 0.8+0.8\cdot 0.8
\end{array}\right]=\left[\begin{array}{ll}
0.2 & 0.8\\
0.2 & 0.8
\end{array}\right]=P$
So, $P^{2}=P.$
$b.$
$P^{4}=P^{2}\cdot P^{2}=P\cdot P=P^{2}=P$
$c.$
$P^{8}=P^{4}\cdot P^{4}=P\cdot P=P^{2}=P$
$d.$
All powers of P equal P, so $P^{1,000}=P$
